NCERT Solutions · Class 12 · Chapter 10: Vector Algebra
NCERT Solutions for Class 12 Maths Chapter 10 Exercise 10.3
Exercise 10.3: Scalar (dot) product. \(\vec a \cdot \vec b = |\vec a||\vec b|\cos\theta = a_1b_1 + a_2b_2 + a_3b_3\). Perpendicular exactly when the dot product is 0. Projection of \(\vec a\) on \(\vec b\): \(\dfrac{\vec a \cdot \vec b}{|\vec b|}\). \(\vec a \cdot \vec a = |\vec a|^2\), so expand \((\vec a + \vec b) \cdot (\vec a + \vec b)\) like an algebraic square.
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Exercise 10.3 questions and solutions
Exercise 10.3, Question 1
Find the angle between \(\vec a\) and \(\vec b\) with \(|\vec a| = \sqrt3\), \(|\vec b| = 2\), \(\vec a \cdot \vec b = \sqrt6\).
Show \(\tfrac17(2\hat i + 3\hat j + 6\hat k)\), \(\tfrac17(3\hat i - 6\hat j + 2\hat k)\), \(\tfrac17(6\hat i + 2\hat j - 3\hat k)\) are unit vectors and mutually perpendicular.
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Each has components with squares summing to \(\tfrac{49}{49} = 1\).
\(\vec a = 2\hat i + 2\hat j + 3\hat k\), \(\vec b = -\hat i + 2\hat j + \hat k\), \(\vec c = 3\hat i + \hat j\), and \(\vec a + \lambda\vec b\) is perpendicular to \(\vec c\). Find \(\lambda\).
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\[\vec a + \lambda\vec b = (2 - \lambda)\hat i + (2 + 2\lambda)\hat j + (3 + \lambda)\hat k\]
Show \(|\vec a|\vec b + |\vec b|\vec a\) is perpendicular to \(|\vec a|\vec b - |\vec b|\vec a\) for non-zero \(\vec a, \vec b\).
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Expand: \[(|\vec a|\vec b + |\vec b|\vec a) \cdot (|\vec a|\vec b - |\vec b|\vec a) = |\vec a|^2\,\vec b \cdot \vec b - |\vec a||\vec b|\,\vec b \cdot \vec a + |\vec b||\vec a|\,\vec a \cdot \vec b - |\vec b|^2\,\vec a \cdot \vec a\]
The middle terms cancel (\[\vec a \cdot \vec b = \vec b \cdot \vec a\]), and \[|\vec a|^2|\vec b|^2 - |\vec b|^2|\vec a|^2 = 0\]; a zero dot product of non-zero vectors means they are perpendicular.
\(\vec a, \vec b, \vec c\) are unit vectors with \(\vec a + \vec b + \vec c = \vec 0\). Find \(\vec a \cdot \vec b + \vec b \cdot \vec c + \vec c \cdot \vec a\).
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\[\begin{aligned}|\vec a + \vec b + \vec c|^2 &= 3 + 2(\vec a \cdot \vec b + \vec b \cdot \vec c + \vec c \cdot \vec a) \\ &= 0\end{aligned}\]
\(\vec a\) is non-zero with magnitude a, \(\lambda\) a non-zero scalar. \(\lambda\vec a\) is a unit vector if: (A) \(\lambda = 1\) (B) \(\lambda = -1\) (C) \(a = |\lambda|\) (D) \(a = \dfrac{1}{|\lambda|}\)
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Vector Algebra has 39 original board-style questions (MCQ, assertion–reason, short and long answers, case studies) with step mark schemes, revision notes and a four-step Route to 95. Three sample questions are open to everyone; a free account opens the rest.
Textbook: NCERT Mathematics Class 12, Parts I and II (rationalised edition, 2023-24 reprint onward), free from ncert.nic.in. Question statements are shortened to the minimum needed; the solutions and tips are our own. CBSE Math Revision is independent and not affiliated with NCERT or CBSE. Spotted a slip? Tell us and it goes in the corrections log.